Construction of the topological toric theory

我们从拓扑学的角度发展了环簇理论。在这几年里，我与Akio教授一起工作，发现环面流形的几何性质可以用称为多扇的组合对象来描述。特别地，我们找到了用与环面流形相联系的多扇来刻画环面流形的椭圆亏格的一个简洁公式，并得到了环面流形的N级椭圆亏格为零的一个定理，即如果流形的第一个陈氏类可被N整除，则作为这个消失定理的推论，如果复数维n的完备环面簇M的第一个陈氏类可被N整除，则N必小于或等于n+1，且当N=n+L时，M同构于复保护空间。这是著名的Kobayashi-Ochiai或Mori定理的环面版本。我邀请莫斯科国立大学的Taras Panov呆了一个月，研究了环面流形M的等变上同调及其轨道空间的上同调。结果表明，当M的上同调环为二次生成时，M的等变上同调环是Stanley-Reisner环，从上同调的角度看，M的轨道空间具有与凸多面体相同的形式。我们还研究了奇次上同调为零的情形。原来，这起案件是通过打击前一起案件而获得的。有趣的是，这种情况下M的等变上同调提供了Stanley-Reisner环的推广。斯坦利在大约十年前就已经提出了这样的环，但我们可能会认为我们的结果给出了环的几何意义。沿着这条线，我证明了Stanley关于Gorenstein*单纯偏序集的h-向量的一个猜想。证明是纯代数的，但这个想法源于拓扑学，这表明组合学、交换代数和拓扑学之间有着密切的联系。

We developed the theory of toric varieties from the topological viewpoint. In these several years I worked with Professor Akio Hattori and found that geometrical properies of a torus manifold can be described in terms of a combinatorial object called a multi-fan. In particular, we found a neat formula describing the elliptic genus of a torus manifold in terms of the multi-fan associated with the torus manifold, and obtained a vanishing theorem saying that the level N elliptic genus of a torus manifold vanishes if the 1st Chern class of the manifold is divisible by N. As a corollary of this vanishing theorem, we obtained a result that if the 1st Chern class of a complete toric variety M of complex dimension n is divisible by N, then N must be less than or equal to n+1, and in case N=n+l, M is isomorphic to the complex protective space. This is a toric version of the famous Kobayashi-Ochiai or Mori's theorem.I invited Taras Panov from Moscow State University for a month and studied the equivariant cohomology of a torus manifold M and the cohomology of its orbit space. As a result, it turned out that when the cohomology ring of M is generated in degree two, the equivariant cohomology of M is a Stanley-Reisner ring and the orbit space of M has the same form as a convex polytope from a cohomological point of view. We also studied the case where M has vanishing odd degree cohomology. It turns out that this case is obtained by blowing down the previous case. Interestingly, the equivariant cohomology of M in this case provides a generalization of the Stanley-Reisner ring. The ring like this was already introduced by Stanley about ten years ago but we may think of our results as giving a geometrical meaning of the ring. Along this line, I proved a conjecture by Stanley about the h-vector of a Gorenstein* simplicial poset. The proof is purely algebraic but the idea stems from topology and this shows a close connection between combinatorics, commutative algebra and topology.